A lithographic apparatus is a machine that applies a desired pattern onto a substrate, usually onto a target portion of the substrate. A lithographic apparatus can be used, for example, in the manufacture of integrated circuits (ICs). In that instance, a patterning device, which is alternatively referred to as a mask or a reticle, may be used to generate a circuit pattern to be formed on an individual layer of the IC. This pattern can be transferred onto a target portion (e.g. comprising part of one, or several dies) on a substrate (e.g. a silicon wafer). Transfer of the pattern is typically via imaging onto a layer of radiation-sensitive material (resist) provided on the substrate. In general, a single substrate will contain a network of adjacent target portions that are successively patterned. Known lithographic apparatus include so-called steppers, in which each target portion is irradiated by exposing an entire pattern onto the target portion at one time, and so-called scanners, in which each target portion is irradiated by scanning the pattern through a radiation beam in a given direction (the “scanning”-direction) while synchronously scanning the substrate parallel or anti-parallel to this direction. It is also possible to transfer the pattern from the patterning device to the substrate by imprinting the pattern onto the substrate.
In order to monitor the lithographic process, it is necessary to measure parameters of the patterned substrate, for example critical linewidth in a developed metrology target. There are various techniques for making measurements of the microscopic structures formed in lithographic processes, including the use of scanning electron microscopes and various specialized tools. A fast and non-invasive form of specialized inspection tool is a scatterometer in which a beam of radiation is directed onto a metrology target (see FIG. 5a) on the surface of the substrate and properties of the scattered or reflected beam are measured (see FIG. 5b). By comparing the properties of the beam before and after it has been reflected or scattered by the substrate, the properties of the metrology target can be determined. This can be done, for example, by comparing the reflected beam with data stored in a library of known measurements associated with known metrology target properties. Two main types of scatterometer are known. Spectroscopic scatterometers direct a broadband radiation beam onto the metrology target and measure the spectrum (intensity as a function of wavelength) of the radiation scattered into a particular narrow angular range. Angularly resolved scatterometers use a monochromatic radiation beam and measure the intensity of the scattered radiation as a function of angle.
The properties of the metrology target can also be determined by modelling the metrology target and scattering by it, and then matching modeled and measured scatterometry results. The modelling may be performed using a numerical solver.
In angle-resolved metrology, the image on the detector in an angularly resolved scatterometer is determined by the outgoing scattered waves. However, in many cases a single outgoing-wave direction can be the result of multiple incident-wave directions (see FIG. 6). For a solver that computes electromagnetic scattering properties such as reflection or transmission coefficients as a function of the direction of incidence, the solver has to be run several times for each angle of incidence that contributes to the direction of scattering. The subsequent step of computing derivatives with respect to structural parameters results in even more computations, especially when finite-difference approximations are used.